Fractions

When the bus is stuck in traffic, or when I’m curled restlessly in bed during the darkest hours of morning when sleep will not come and all the old wounds on my heart ache like they were newly made, I sometimes think back over my education and add up all the time my teachers wasted. The worst, perhaps, was the mathematics, for which entire years of schooling went for naught. “Pre-Calculus”? Pfft. AP Computer Science? Pfft++. All in all, I’d say that upwards of a third of my mathematics schooling before university was a waste of time, and another third was so incompetently done that any student who hadn’t already been hooked on science and learning would have been completely sunk.

So, I find it easy to sympathize with people who say that math education needs a severe overhaul. I’m willing to contemplate big curriculum changes, but of course, you have to convince me that the specific changes you have in mind will actually do any good. When a proposal comes down the wire to eliminate fractions, I reserve the right to chortle and guffaw.
The plan in question comes from Dennis DeTurck, who says that we should scrap long division, the extraction of square roots by hand, manual multiplication of “large numbers” and, as I said, fractions. In fact, he suggests we hold off teaching fractions until after the student learns calculus.

DeTurck dismisses all criticism leveled at his proposal:

There were blogs and rants, and there were some critical e-mails, they’d always boil down to: ‘What would we do in cooking and carpentry?’

The problem is more fundamental than that. First of all, if you don’t teach fractions in elementary school, your kids are going to have a real problem when they reach junior high and try to buy marijuana from their elders.

“Hey, man, how much is an eighth?”

“How much cash you got on ya?”

“No, man, how much is an eighth?”

Second, I’ve thought about it, and the only thing I can see that abolishing fractions will do is make teaching division harder. And perish the idea of understanding repeating decimals or irrational numbers. In tenth grade, we were finally shown a proof that the square root of two is irrational (I say “finally” because I’d read about it six years earlier). Our algebra teacher didn’t prove directly that the decimal expansion of [tex]\sqrt{2}[/tex] continued infinitely without repetition,

[tex]\sqrt{2} = 1.414213562373\ldots,[/tex]

but instead she showed that [tex]\sqrt{2}[/tex] could not be written as the ratio of two integers, no matter how large they were. If you didn’t have a solid grounding in fractions, would this approach even occur to you, and how easily could you follow the proof when it was presented on the blackboard?

If your idea of math is to put some numbers in and get a number out, then DeTurck may be right…as long as you have a calculator (try doing 9.33*.429 in your head…now try (28/3)*(3/7)). But you can’t even do simple algebra without a fairly good understanding of fractions…much less trigonometry or calculus!

I remember infuriating my grade-school teachers by not writing down my carries when doing arithmetic by hand. Tricks I learned from my mother, or from reading Feynman’s storybooks, received an even worse reception. For example, how could I possibly explain that I didn’t need to go the long way round to square a number near 50, when I knew that 50 squared was 2500, and I could exploit the identity

[tex](a + b)^2 = a^2 + 2ab + b^2.[/tex]

The idea that multiple methods exist for finding the right answer appeared to be entirely foreign to the math teacher’s mind. Somehow, I survived.

I also recall a great many long division problems. Perhaps we did have too many of those, although I have the sneaky feeling that what we really needed was not to reduce everyone’s quota, but to find a mechanism whereby the students who picked up the material more quickly could move ahead, and those who needed more time could be given it. It would also have been nice to see better attempts to make math useful and applicable to the real world, perhaps by introducing basic probability theory or something of the sort.

Incidentally, I was never taught how to take square roots by hand. (Performing a marginally scientific poll, I find that nobody in the office did either.) I picked it up, somewhere along the line, but it was never part of my formal curriculum. In this case, DeTurck might be attacking a target which isn’t there.

UPDATE:Mark Chu-Carroll has written his own statement on DeTurck’s antics (because everybody and their brother sends news stories about stupid math to the master of the Good Math, Bad Math blog). He writes,

The biggest problem, in my experience, with how math is taught is that we focus on mechanics to the exclusion of understanding. Switching to pure decimals without fractions is carrying that to a ridiculous extreme.

I would be cautious about picking one problem as “the biggest,” but I’m a grumpy guy. This is, indeed, a very real problem: how many of us were taught why long division works? I know I wasn’t.

UPDATE (3 February): As is often the case, I find myself agreeing with Jason Rosenhouse pretty much all the way.

I was taught to do square roots by hand… But as usual for me, I learned it from my father. I rarely learned anything about math in school – my dad always taught it to me for fun long before the school got around to it.

Are you saying computer science in high school was a waste of time!? I am deeply and permanently offended by such an accusation. I never would’ve been able to come up with a computer program that happened to store all of Kim and my inside jokes that year had it not been for the infinite wisdom of what’s her face (Naussauc? Nosack? I remember that’s how she pronounced it).
On another note, I think I’ve forgotten everything I ever learned about math except simple arithmetic. And even that I fuck up sometimes. Go go law school!

Actually I did learn how to take square roots by hand with an algorithm that resembled division. Perhaps I had a good teacher or perhaps it is just because I belong to the generation that latter some of us at least had to learn how to use a slide rule. But regardless years latter I had forgotten it and reconstructed it taking advantage of the Binomial Theorem, eliminating all but the numbers you end up with the same algorithm I was taught in grade school. But the algebra reveals that this algorithm can be trivially generalized to nth order roots.

The key is, are we trying to get students to understand or just to get answers?

I was astonished that my daughter, in 6th grade, didn’t understand what the decimal places mean, IOW powers of ten. How can you understand math if you don’t know what the numbers mean?

I teach low-level chemistry. When I do percent composition, the kids don’t understand that “percent” means the same as “parts per hundred.” They plug numbers into the calculators and get an answer, and have no idea whether it it makes sense. They will tell me that a copper containing compound is 450% copper, and don’t realize that is a ridiculous answer.

I think that is one of the worst aspects of focusing on year-end testing. The emphasis on getting an answer rather than understanding.

I have a two stage solution: First, require that all teachers, including pre-k, have a year of calculus and a year of physics. Second, take all of the calculators in the elementary schools and middle schools and beat them into little pieces.

What really upsets me is that there is now an “Electronic Banking Monopoly” game . . . uses a debit card rather than cash. That is just so WRONG. Paying for properties and making change out of the bank taught GENERATIONS of children how to add and subtract without a piece of paper.

And yes, I did learn how to take square roots by hand . . . in elementary school if you can believe it (of course it was an extension of all that long division training that we did) . . . in the 1960s . . . in the time before calculators and when we were taught all sorts of tricks for mental math. Anecdote . . . we gave your maternal grandfather a little five function calculator back in the late 70s. It came with a little book of “calculator tricks you can do.” He would check the calculator’s answers . . . didn’t trust it. Of course, this is the man who divided fractions in his head. You have a lot of Grandma Stacey in you . . . but a bunch of Grandad Chilton, too.

I sat in an Advanced Project Management class last fall and watched people yank out calculators to do some “earned value” problems (simple algebra equation stuff) and we’re giving ludicrous answers (magnitudes off) because of decimal issues. These are folks with college degrees . . . managing projects. So scary. There was no “intuitiveness.” I didn’t even take a calculator to class and just turned my paper over and did the multiplication and division the old way. I could even “simplify” before I started my division!

the only thing I can see that abolishing fractions will do is make teaching division harder.

Is this a joke? At the very least, it seems like quite an understatement. It seems abolishing fractions means abolishing division as well. After all, what’s the difference between the fraction 3/2 and 3 divided by 2?

Is Dennis DeTurck suggesting that we restrict attention to integers? Or maybe he accepts decimals like 1.5, but thinks we shouldn’t explain what they mean?

I haven’t had time to read his opinions, and I’m not sure I should find time.

BaldApe said: “I have a two stage solution: First, require that all teachers, including pre-k, have a year of calculus and a year of physics. Second, take all of the calculators in the elementary schools and middle schools and beat them into little pieces.”

I like these ideas. Too often teachers of lower level mathematics courses ASSUME that they are helping prepare students for higher level courses by teaching them calculator skills. I have addressed this issue with some of them in the past and they tell me I am the one that needs to change. What is wrong with these people? They are creating “happy idiots” who know very little mathematics when they get to my classes and are now required to math without a calculator and learn the skills that they were not taught when they should have been. Maybe all math teachers should also be required to teach a year of Calculus so that they know where the students are heading. After this type of experience they may not be as quick to substitute the calcuator for skills they should be teaching.

It’s funny, but until this topic came up, I hadn’t really thought about how seldom I actually use a calculator! When paying the bill at a restaurant, I can compute the tip in my head (the challenge arises when multiple people have to pool money and nobody has quite the right amount of cash. . .). I used a calculator in high school, but it died while I was at university and I never bothered to buy a replacement: homework and exam problems had become about the manipulation of symbols, while the bits of arithmetic which were still necessary could be done on scratch paper.

If a whole lot of number-crunching is needed, I’ll bet euros to dollars you’re on a computer anyway, and understanding the symbolic manipulations of a computer language is, again, essential.

A “no calculators” rule sounds to me like slapping a Band-Aid on the problem, one which won’t stick very well anyway. My hunch is that we should be transitioning the whole curriculum over to problems where calculators just aren’t that helpful. That said, all the different ways one can destroy calculators — liquid nitrogen, trebuchet, etc. — might themselves offer “teachable moments.”